The word is chosen by analogy with topos which can be regarded as “a good place to do set theory,” but there are notable differences between the two situations; a more direct categorification of a topos is, unsurprisingly, a 2-topos. In contrast, cosmoi also include enriched category theory, while toposes do not allow non-cartesian enrichment.
There are a number of different, inequivalent, definitions of “cosmos” in the literature.
Ross Street has taken a different tack, defining a “cosmos” to be the collection of (enriched) categories and relevant structure for doing category theory, rather than the “base” category over which the enrichment occurs.
In his paper “Elementary cosmoi,” Street defined a (fibrational) cosmos to be a 2-category in which internal fibrations are well-behaved and representable by a structure of “presheaf objects” (later realized to be a special sort of Yoneda structure?). Note that while this includes , it does not include - for non-cartesian , since internal fibrations are poorly behaved there.
In his paper “Cauchy characterization of enriched categories,” Street instead defined a “cosmos” to be a 2-category that behaves like the 2-category - of enriched categories and profunctors. The precise definition: a cosmos is a 2-category (or bicategory) such that: * Small (weak, or bi-) coproducts exist. * Each monad admits a Kleisli construction? (analogous to the exactness of a topos). * It is locally small-cocomplete, i.e. its hom-categories have small colimits that are preserved by composition on each side. * There exists a small “Cauchy generator”.
These hypotheses imply that it is equivalent to the bicategory of categories and profunctors enriched over some “base” bicategory. (Note the generalization from enrichment over a monoidal category to enrichment over a bicategory.)